Consider The Incomplete Paragraph Proof.Given: Isosceles Right Triangle XYZ \[$(45^{\circ}-45^{\circ}-90^{\circ}\$\] Triangle)Prove: In A \[$45^{\circ}-45^{\circ}-90^{\circ}\$\] Triangle, The Hypotenuse Is \[$\sqrt{2}\$\] Times

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The Incomplete Paragraph Proof: A Step-by-Step Guide to Proving the Hypotenuse of an Isosceles Right Triangle

In mathematics, a right triangle is a fundamental concept that has been studied for centuries. One of the most interesting types of right triangles is the isosceles right triangle, also known as a 45°-45°-90° triangle. This type of triangle has two equal sides and a right angle, making it a unique and fascinating shape. In this article, we will explore the concept of an isosceles right triangle and prove that the hypotenuse is √2 times the length of one of its legs.

What is an Isosceles Right Triangle?

An isosceles right triangle is a type of right triangle that has two equal sides and a right angle. The two equal sides are called the legs, and the side opposite the right angle is called the hypotenuse. In an isosceles right triangle, the two legs are equal in length, and the hypotenuse is the longest side. The angles of an isosceles right triangle are always 45°-45°-90°, with the two 45° angles being equal.

The Incomplete Paragraph Proof

Given: Isosceles right triangle XYZ (45°-45°-90° triangle)

Prove: In a 45°-45°-90° triangle, the hypotenuse is √2 times the length of one of its legs.

To prove this statement, we need to use the properties of isosceles right triangles and the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Step 1: Draw a Diagram

Let's draw a diagram of an isosceles right triangle XYZ.

  X
 / \
/   \
Y---Z

In this diagram, X and Y are the two equal legs, and Z is the hypotenuse.

Step 2: Label the Diagram

Let's label the diagram with the lengths of the sides.

  X (a)
 / \
/   \
Y (a)---Z (b)

In this diagram, a is the length of one of the legs, and b is the length of the hypotenuse.

Step 3: Apply the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. In this case, we can write:

b² = a² + a²

Simplifying the equation, we get:

b² = 2a²

Step 4: Take the Square Root

To find the length of the hypotenuse, we need to take the square root of both sides of the equation.

b = √(2a²)

Simplifying the equation, we get:

b = √2a

In this article, we have proven that in an isosceles right triangle, the hypotenuse is √2 times the length of one of its legs. This is a fundamental property of isosceles right triangles and is used in many mathematical applications. We have used the Pythagorean theorem and the properties of isosceles right triangles to prove this statement.

Real-World Applications

The concept of an isosceles right triangle has many real-world applications. For example, in construction, architects use isosceles right triangles to design buildings and bridges. In engineering, isosceles right triangles are used to calculate the stresses and strains on materials. In physics, isosceles right triangles are used to describe the motion of objects.

In conclusion, the concept of an isosceles right triangle is a fundamental concept in mathematics that has many real-world applications. We have proven that in an isosceles right triangle, the hypotenuse is √2 times the length of one of its legs. This is a fundamental property of isosceles right triangles and is used in many mathematical applications.

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  • "Trigonometry" by I.M. Gelfand
  • "Mathematics for the Nonmathematician" by Morris Kline
  • "The Pythagorean Theorem" by Alfred S. Posamentier
  • "Geometry: A Comprehensive Introduction" by Dan Pedoe
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    Frequently Asked Questions: Isosceles Right Triangles

In our previous article, we explored the concept of an isosceles right triangle and proved that the hypotenuse is √2 times the length of one of its legs. In this article, we will answer some of the most frequently asked questions about isosceles right triangles.

Q: What is an isosceles right triangle?

A: An isosceles right triangle is a type of right triangle that has two equal sides and a right angle. The two equal sides are called the legs, and the side opposite the right angle is called the hypotenuse.

Q: What are the angles of an isosceles right triangle?

A: The angles of an isosceles right triangle are always 45°-45°-90°, with the two 45° angles being equal.

Q: How do I prove that the hypotenuse of an isosceles right triangle is √2 times the length of one of its legs?

A: To prove this statement, you can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. In an isosceles right triangle, the two legs are equal in length, so you can write:

b² = a² + a²

Simplifying the equation, you get:

b² = 2a²

Taking the square root of both sides, you get:

b = √2a

Q: What are some real-world applications of isosceles right triangles?

A: Isosceles right triangles have many real-world applications. For example, in construction, architects use isosceles right triangles to design buildings and bridges. In engineering, isosceles right triangles are used to calculate the stresses and strains on materials. In physics, isosceles right triangles are used to describe the motion of objects.

Q: Can I use isosceles right triangles to solve problems in other areas of mathematics?

A: Yes, isosceles right triangles can be used to solve problems in other areas of mathematics, such as trigonometry and geometry. For example, you can use isosceles right triangles to calculate the lengths of sides of triangles and to solve problems involving right triangles.

Q: How do I draw an isosceles right triangle?

A: To draw an isosceles right triangle, you can start by drawing a line segment to represent one of the legs. Then, you can draw a line segment perpendicular to the first line segment to represent the other leg. Finally, you can draw a line segment connecting the two legs to represent the hypotenuse.

Q: What are some common mistakes to avoid when working with isosceles right triangles?

A: Some common mistakes to avoid when working with isosceles right triangles include:

  • Assuming that the two legs are equal in length without checking
  • Failing to use the Pythagorean theorem to calculate the length of the hypotenuse
  • Drawing the triangle incorrectly, such as drawing the hypotenuse as one of the legs

In this article, we have answered some of the most frequently asked questions about isosceles right triangles. We have covered topics such as the definition of an isosceles right triangle, the angles of an isosceles right triangle, and the real-world applications of isosceles right triangles. We have also provided some tips and tricks for working with isosceles right triangles and avoiding common mistakes.

  • "Geometry" by Michael S. Klamkin
  • "Trigonometry" by I.M. Gelfand
  • "Mathematics for the Nonmathematician" by Morris Kline
  • "The Pythagorean Theorem" by Alfred S. Posamentier
  • "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • "Trigonometry: A First Course" by I.M. Gelfand